Abstract: I will give an overview of the mathematics of quantum many-body systems. The main difference with single-particle systems is that many-body Hamiltonians are not short-range (their kernels are not band matrices in the position basis). As a consequence, functions of local Hamiltonians are no longer local. Instead, people have relied on propagation bounds and mean-field theories to obtain spectral information. I will detail a few classical results and give some new applications.